Fig. 1: In coffee percolation, soluble compounds leave the coffee grounds and join the water to form coffee. Insoluble compounds (and granulates) remain within the coffee filter.
Fig. 2: Percolation in a square lattice (Click to animate)

In physics, chemistry and materials science, percolation (from Latin percolare, "to filter" or "trickle through") refers to the movement and filtering of fluids through porous materials. It is described by Darcy's law. Broader applications have since been developed that cover connectivity of many systems modeled as lattices or graphs, analogous to connectivity of lattice components in the filtration problem that modulates capacity for percolation.


During the last decades, percolation theory, the mathematical study of percolation, has brought new understanding and techniques to a broad range of topics in physics, materials science, complex networks, epidemiology, and other fields. For example, in geology, percolation refers to filtration of water through soil and permeable rocks. The water flows to recharge the groundwater in the water table and aquifers. In places where infiltration basins or septic drain fields are planned to dispose of substantial amounts of water, a percolation test is needed beforehand to determine whether the intended structure is likely to succeed or fail. In two dimensional square lattice percolation is defined as follows. A site is "occupied" with probability p or "empty" (in which case its edges are removed) with probability 1 – p; the corresponding problem is called site percolation, see Fig. 2.

Percolation typically exhibits universality. Statistical physics concepts such as scaling theory, renormalization, phase transition, critical phenomena and fractals are used to characterize percolation properties. Combinatorics is commonly employed to study percolation thresholds.

Due to the complexity involved in obtaining exact results from analytical models of percolation, computer simulations are typically used. The current fastest algorithm for percolation was published in 2000 by Mark Newman and Robert Ziff.[1]


  • Coffee percolation (see Fig. 1), where the solvent is water, the permeable substance is the coffee grounds, and the soluble constituents are the chemical compounds that give coffee its color, taste, and aroma.
  • Movement of weathered material down on a slope under the earth's surface.
  • Cracking of trees with the presence of two conditions, sunlight and under the influence of pressure.
  • Collapse and robustness of biological virus shells to random subunit removal (experimentally-verified fragmentation of viruses).[2][3][4]
  • Robustness of networks to random and targeted attacks.[5]
  • Transport in porous media.
  • [].[6][7][8]
  • Surface roughening.[citation needed]
  • Dental percolation, increase rate of decay under crowns because of a conducive environment for strep mutants and lactobacillus
  • Potential sites for septic systems are tested by the "perk test". Example/tepiheory: A hole (usually 6–10 inches in diameter) is dug in the ground surface (usually 12–24" deep). Water is filled in to the hole, and the time is measured for a drop of one inch in the water surface. If the water surface quickly drops, as usually seen in poorly-graded sands, then it is a potentially good place for a septic "leach field". If the hydraulic conductivity of the site is low (usually in clayey and loamy soils), then the site is undesirable.
  • Traffic percolation.[9][10]
  • Percolation in the presence of reinforced (decentralization of the network) has been studied by Yuan et al.[11]

See also


  1. ^ Newman, Mark; Ziff, Robert (2000). "Efficient Monte Carlo Algorithm and High-Precision Results for Percolation". Physical Review Letters. 85 (19): 4104–4107. arXiv:cond-mat/0005264. Bibcode:2000PhRvL..85.4104N. CiteSeerX doi:10.1103/PhysRevLett.85.4104. PMID 11056635. S2CID 747665.
  2. ^ Brunk, Nicholas E.; Twarock, Reidun (2021-07-23). "Percolation Theory Reveals Biophysical Properties of Virus-like Particles". ACS Nano. American Chemical Society (ACS). 15 (8): 12988–12995. doi:10.1021/acsnano.1c01882. ISSN 1936-0851. PMC 8397427. PMID 34296852.
  3. ^ Brunk, Nicholas E.; Lee, Lye Siang; Glazier, James A.; Butske, William; Zlotnick, Adam (2018). "Molecular jenga: The percolation phase transition (collapse) in virus capsids". Physical Biology. 15 (5): 056005. Bibcode:2018PhBio..15e6005B. doi:10.1088/1478-3975/aac194. PMC 6004236. PMID 29714713.
  4. ^ Lee, Lye Siang; Brunk, Nicholas; Haywood, Daniel G.; Keifer, David; Pierson, Elizabeth; Kondylis, Panagiotis; Wang, Joseph Che-Yen; Jacobson, Stephen C.; Jarrold, Martin F.; Zlotnick, Adam (2017). "A molecular breadboard: Removal and replacement of subunits in a hepatitis B virus capsid". Protein Science. 26 (11): 2170–2180. doi:10.1002/pro.3265. PMC 5654856. PMID 28795465.
  5. ^ R. Cohen and S. Havlin (2010). "Complex Networks: Structure, Robustness and Function". Cambridge University Press.
  6. ^ Parshani, Roni; Carmi, Shai; Havlin, Shlomo (2010). "Epidemic Threshold for the Susceptible-Infectious-Susceptible Model on Random Networks". Physical Review Letters. 104 (25): 258701. arXiv:0909.3811. Bibcode:2010PhRvL.104y8701P. doi:10.1103/PhysRevLett.104.258701. ISSN 0031-9007. PMID 20867419.
  7. ^ Grassberger, Peter (1983). "On the Critical Behavior of the General Epidemic Process and Dynamical Percolation". Mathematical Biosciences. 63 (2): 157–172. doi:10.1016/0025-5564(82)90036-0.
  8. ^ Newman, M. E. J. (2002). "Spread of epidemic disease on networks". Physical Review E. 66 (1 Pt 2): 016128. arXiv:cond-mat/0205009. Bibcode:2002PhRvE..66a6128N. doi:10.1103/PhysRevE.66.016128. PMID 12241447. S2CID 15291065.
  9. ^ D. Li, B. Fu, Y. Wang, G. Lu, Y. Berezin, H.E. Stanley, S. Havlin (2015). "Percolation transition in dynamical traffic network with evolving critical bottlenecks". PNAS. 112 (3): 669–72. Bibcode:2015PNAS..112..669L. doi:10.1073/pnas.1419185112. PMC 4311803. PMID 25552558.CS1 maint: uses authors parameter (link)
  10. ^ Guanwen Zeng, Daqing Li, Shengmin Guo, Liang Gao, Ziyou Gao, HEugene Stanley, Shlomo Havlin (2019). "Switch between critical percolation modes in city traffic dynamics". Proceedings of the National Academy of Sciences. 116 (1): 23–28. Bibcode:2019PNAS..116...23Z. doi:10.1073/pnas.1801545116. PMC 6320510. PMID 30591562.CS1 maint: uses authors parameter (link)
  11. ^ X. Yuan, Y. Hu, H.E. Stanley, S. Havlin (2017). "Eradicating catastrophic collapse in interdependent networks via reinforced nodes". PNAS. 114 (13): 3311–3315. arXiv:1605.04217. Bibcode:2017PNAS..114.3311Y. doi:10.1073/pnas.1621369114. PMC 5380073. PMID 28289204.CS1 maint: uses authors parameter (link)

Further reading

  • Kesten, Harry; "What is percolation?", in Notices of the AMS, May 2006.
  • Sahimi, Muhammad; Applications of Percolation Theory, Taylor & Francis, 1994. ISBN 0-7484-0075-3 (cloth), ISBN 0-7484-0076-1 (paper).
  • Grimmett, Geoffrey; Percolation (2. ed). Springer Verlag, 1999.
  • Stauffer, Dietrich ; and Aharony, Ammon; Introduction to Percolation Theory, Taylor & Francis, 1994, revised second edition, ISBN 9780748402533.
  • Bunde, Armin; Havlin, Shlomo (editors); Fractals and Disordered Systems, Springer, 1996.
  • Kirkpatrick, Scott; "Percolation and Conduction", in Reviews of Modern Physics, 45, 574, 1973.
  • Ben-Avraham, Daniel; Havlin, Shlomo; Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, 2000.
  • Rodrigues, Edouard; Remarkable properties of pawns on a hexboard
  • Cohen, Reuven; and Havlin, Shlomo; Complex Networks: Structure, Robustness and Function, Cambridge University Press, 2010, ISBN 978-0521841566.
  • Bollobás, Béla; Riordan, Oliver; Percolation, Cambridge University Press, 2006, ISBN 0521872324.
  • Grimmett, Geoffrey; Percolation, Springer, 1999

External links

  • Introduction to Percolation Theory: short course by Shlomo Havlin